Well-posedness and porosity in best approximation problems
In general a closed subset \(A\) of a Banach space \(X\) does not contain a best approximation to a point \(x\in X-A\). However, there are Banach spaces in which the set of such points,\(x\), which have best approximations in \(A\), form a dense subset (or more) of \(X\). The development of the field includes proving equivalent properties of the Banach spaces, determining the size of the set of points that have best approximations, and considering uniqueness of best approximations and the convergence of minimizing sequences. This work advances the theory by changing the framework to allow the set \(A\) to vary. The results apply to an arbitrary Banach space \(X\). There are three types of results. We describe simple forms of two of them. The first fixes the point \(x\), and defines a complete metric on the set \(S\) of closed nonempty subsets of \(X\). The second, extends this metric to \(S\times X\). The authors show that, in both metric spaces, the points that represent a setting with a unique best approximation form a dense set (in fact a set with a \(\sigma\)-porous complement).
scientific article; zbMATH DE number 1799630